Summary of Visualizing the sphere and the hyperbolic plane: five projections of each

00:00:00 - 00:10:00

This video explains five projections of the sphere and the hyperbolic plane. The azimuthal equidistant projection preserves hyperbolic area, while the Lambert azimuthal equal-area projection is the unique projection that is azimuthal and preserves hyperbolic area. The pseudorthographic projection maps the hyperbolic plane to the whole Euclidean plane, while the orthographic and gnomonic projections map the sphere to the orthographic and gnomonic projections, respectively.

• 00:00:00 This video explains five different projections of the sphere to the plane, and five analogous projections of the hyperbolic plane. The Orthographic projection corresponds to viewing the sphere from infinitely far away, so that only one hemisphere is visible. The Gnomonic projection preserves alignment, while the Stereographic projection preserves circles and preserves areas. The Lambert azimuthal equal-area projection preserves both angles and areas.
• 00:05:00 The five projections of the sphere and the hyperbolic plane are shown, starting with the azimuthal equidistant projection and proceeding to the Poincaré disk model, the gnomonic projection of the sphere, and the orthographic projection. The hyperbolic plane cannot be easily embedded in Euclidean space, so the Poincaré disc model is a good one to start. The lines of the hyperbolic plane in the Poincaré disc projection correspond to Euclidean arcs of circle which are orthogonal to the circle at infinity at both edges. The figure we have here is a regular tiling of the hyperbolic plane by regular heptagons all the same size. The overall movement we see is a hyperbolic translation: in the spherical case there was no difference between rotation and translation, a rotation of the sphere around a polar axis is the same as a translation along the corresponding equator. In the ordinary Euclidean plane translations and rotations are different, but the translations are merely a limit case of rotations when the center goes to infinity and the angle become smaller. In hyperbolic geometry translations and rotations are truly different: and you can tell that this is a translation because it preserves
• 00:10:00 The visual representations of the sphere and the hyperbolic plane are shown in this video, along with five projections of each. The pseudorthographic projection maps the hyperbolic plane to the whole Euclidean plane, while the orthographic and gnomonic projections map the sphere to the orthographic and gnomonic projections, respectively. The azimuthal equidistant projection preserves hyperbolic area, and the Lambert azimuthal equal-area projection is the unique projection that is azimuthal and preserves hyperbolic area. Every heptagon has the same size as all the others, and is projected to a region of equal area in the azimuthal equidistant projection.